Chapter 2 Intelligent Behaviour Modelling and Control for ...

Chapter 2Intelligent Behaviour Modelling and Controlfor Mobile Manipulators

Ayssam Elkady, Mohammed Mohammed, Eslam Gebriel, and Tarek Sobh

Abstract In the last several years, mobile manipulators have been increasinglyutilized and developed from a theoretical viewpoint as well as for practical ap-plications in space, underwater, construction and service environments. The workpresented in this chapter deals with the problem of intelligent behaviour modellingand control of a mobile manipulator for the purpose of simultaneously followingdesired end-effector and platform trajectories. Our mobile manipulator compriseda manipulator arm mounted on a motorized mobile base wheelchair. The need foraccurate modelling of the mobile manipulator is crucial in designing and controllingthe motion of the robot to achieve the target precision and manipulability require-ments. In this chapter, we propose a new method for measuring the manipulabilityindex used for serial manipulators. Furthermore, we provide some simulations thatare implemented on different serial manipulators, such as the Puma 560 manipula-tor, a six degrees of freedom (DOF) manipulator and the Mitsubishi Movemastermanipulator. We then extend the manipulability concept commonly used for serialmanipulators to general mobile manipulator systems.

Keywords Mobile manipulator � Manipulability � Jacobian � Dexterity � Singularvalue decomposition � Singularities � Nonholonomic � Kinematics

2.1 Introduction

Studying the performance characteristics of the robot, such as dexterity, manip-ulability and accuracy, is very important to the design and analysis of a robotmanipulator. The manipulability is the ability to move in arbitrary directions, whilethe accuracy is a measure of how close the manipulator can return to a previouslytaught point. The workspace of a manipulator is a total volume swiped out bythe end effector when it executes all possible motions. The workspace is subdi-vided into the reachable workspace and the dexterous workspace. The reachable

A. Elkady (�)University of Bridgeport, Bridgeport, CT 06604, USAe-mail: [email protected]

A.-E. Hassanien et al. (eds.), Pervasive Computing: Innovations in IntelligentMultimedia and Applications, Computer Communications and Networks,DOI 10.1007/978-1-84882-599-4 2, c� Springer-Verlag London Limited 2009

29

[email protected]

30 A. Elkady et al.

workspace is all point reachable by the end-effector. But the dexterous workspaceconsists of all points that the end-effector can reach with an arbitrary orientationof the end-effector. Therefore, the dexterous workspace is a subset of the reachableworkspace. The dexterity index is a measure of a manipulator to achieve differentorientations for each point within the workspace.

In this chapter, we present a new method for measuring the manipulability index,and some simulations are implemented on different manipulators, such as the Puma560 manipulator, a six degrees of freedom (DOF) manipulator and the MitsubishiMovemaster manipulator. In addition, we describe how the manipulability measureis crucial in performing intelligent behaviour tasks. The manipulability index is con-sidered as a quantitative and performance measure of the ability for realizing sometasks. This measure should be taken into consideration in the design phase of aserial robot and also in the design of control algorithms. Furthermore, we use theproposed method for measuring the manipulability index in serial manipulators togeneralize the standard definition of the manipulability index in the case of mobilemanipulators.

2.2 Prior Work

Klein and Blaho [6] proposed some measures for the dexterity of manipulators,then they compared several measures for the problems of finding an optimal configu-ration for a given end-effector position, finding an optimal workpoint and designingthe optimal link lengths of an arm. They considered four measures for dexterity:determinant, condition number, minimum singular value of the Jacobian and jointrange availability. Salisbury and Craig [8] illustrated hand designs with particularmobility properties. In addition, they gave a definition of accuracy points within ma-nipulator workspace. They used another performance index which is the conditionnumber of the Jacobian. Yoshikawa [13] gave one of the first mathematical measuresfor the manipulability of any serial robot by discussing the manipulating ability ofrobotic mechanisms in positioning and orienting end-effectors. He introduced theterm manipulability, which involves the Jacobian and its transpose; then the evalua-tion of the determinant of the Jacobian can be used to determine the manipulabilitymeasure. Gosselin [3] presented two dexterity indices for planar manipulations, thefirst one is based on a redundant formulation of the velocity equations and the sec-ond one is based on the minimum number of parameters. Then the correspondingindices were derived for spatial manipulators. These indices are based on the condi-tion number of the Jacobian matrix of the manipulators. He considered the dexterityindex, manipulability, condition number and minimum singular value, then he ap-plied these indexes to a SCARA type robot. Van den Doel and Pai [12] introduceda performance measure of robot manipulators in a unified framework based on dif-ferential geometry. The measures are applied to the analysis of two- and three-linkplanar arm. In [5], the authors demonstrated that manipulability of a mechanismis independent of task space coordinates. Furthermore, they provided a proof ofthe independency of the manipulability index on the first DOF. In [7], the author

2 Intelligent Behaviour Modelling and Control for Mobile Manipulators 31

examined two geometric tools for measuring the dexterousness of robot manipula-tors, manipulability ellipsoids and manipulability polytopes. He illustrated that themanipulability ellipsoid does not transform the exact joint velocity constraints intotask space and so may fail to give exact dexterousness measure and optimal directionof motion in task space. Furthermore, he proposed a practical polytope method thatcan be applied to general 6-dimensional task space.

In [10], Sobh and Toundykov presented a prototyping software tool which runsunder the mathematica environment and automatically computes possible optimalparameters of robot arms by applying numerical optimization techniques to the ma-nipulability function, combined with distances to the targets and restrictions on thedimensions of the robot.

Nearly all of the above techniques start by getting the forward kinematics, thenthe Jacobian equation, which relates to the velocity of the end-effector and the jointvelocities.

2.3 Manipulability Measure

2.3.1 Jacobian Matrix

The Jacobian matrix provides a transformation from the velocity of the end-effectorin cartesian space to the actuated joint velocities as shown in equation (2.1)

Px D J Pq; (2.1)

where Pq is an m-dimensional vector that represents a set of actuated joint rates, Pxis an n-dimensional output velocity vector of the end-effector and J is the m � nJacobian matrix. It is possible that m ¤ n. As an example, a redundant manipulatorcan have more than six actuated joints, while the end-effector will at most have sixDOF, so that m > n.

In the singular position, the Jacobian matrix J looses rank. This means that theend-effector looses one or more degrees of twist freedom (i.e., instantaneously, theend-effector cannot move in these directions). The mathematical discussion of sin-gularities relies on the rank of the Jacobian matrix J, which, for a serial manipulatorwith n joints, is a 6 � n matrix. For a square Jacobian, det.J / D 0 is a necessaryand sufficient condition for a singularity to appear.

2.3.2 Singular Value Decomposition Method

The singular value decomposition (SVD) method works for all possible kinematicstructures (i.e., with every Jacobian matrix J with arbitrary dimensions m � n). TheSVD decomposition of any matrix J is on the form:

JmnD Umm˙mnVtnn; (2.2)

32 A. Elkady et al.

with

˙ D

0

BBBBBBBBBB@

�1 0 0 : : : 0 0 : : : 0

0 �2 0 : : : 0 0 : : : 0

0 0: : : 0 : : : 0 : : : 0

0 0 : : : �m�1 0 0 : : : 0

0 0 : : : 0 �m 0 : : : 0

1

CCCCCCCCCCA

:

Such that U and V are orthogonal matrices. Thus,

U tU D Imm; (2.3)

V tV D I nn; (2.4)

where I is the identity matrix and the singular values are in descending orders �1 �

�2 � � � � � �m. The matrix has a zero determinant and is, therefore, singular (it hasno inverse). The matrix has two identical rows (or two identical columns). In otherwords, the rows are not independent. If one row is a multiple of another, then theyare not independent, and the determinant is zero. (Equivalently: If one column is amultiple of another, then they are not independent, and the determinant is zero.) Therank of a matrix is the maximum number of independent rows (or the maximumnumber of independent columns). A square matrix Ann is nonsingular only if itsrank is equal to n. Mathematically, matrix J having a full rank means that the rankof J Dm. In this case, �m¤ 0. When �m 0, the matrix J does not have a full rank,which means that the matrix J looses one or more DOF. This happens physically,when the serial robot has two joint axes coinciding on each other.

2.3.3 Manipulability Measures

Yoshikawa [13] defined the manipulability measure � as the square root of thedeterminant of the product of the manipulator Jacobian by its transpose

� D Œdet.J � J t /�1=2: (2.5)

If the Jacobian matrix J is a square matrix, the manipulability � is equal to the ab-solute value of the determinant of the Jacobian. Using the SVD, the manipulabilitycan be written as follows:

� D �1�2 : : : �m: (2.6)

Another method for the manipulability measure is the reciprocal of the conditionnumber [termed the conditioning index] that was used in [11].


2.3.4 Optimizing the Manipulability Index of Serial ManipulatorsUsing the SVD Method

Our current work addresses the manipulability index for every point within theworkspace of some serial manipulators. The method provided promising results,since it is considered one of the crucial tasks required for designing trajectories oravoiding singular configurations. We propose a new method for measuring the ma-nipulability, then we implemented simulations supporting our method on the Puma560 manipulator, a six DOF manipulator and the Mitsubishi Movemaster manip-ulator. As mentioned in [11], the determinant of a Jacobian cannot be used forexpressing the manipulability’s index. It reaches zero when a manipulator reachesany singular configuration. Another method has been proposed, labelled the recip-rocal of the Jacobian (as in [11]). In past researches, there was an argument aboutwhether the minimum value of the � ’s in (2.2) or the multiplication of all � ’s exactlyrepresents the manipulability’s index [3].

In this work, we propose a new concept for measuring this index, then justifythis concept by visualizing the bands of this index, resulting from our experiments.Moreover, a new relationship between the minimum rank of the Jacobian matrixand the order of one of these � ’s (in (2.2)) can exactly express the manipulability’sindex.

2.3.4.1 The Puma 560 Manipulator: A Case Study

In case of the singular configuration of the Puma 560 manipulator atQ D Œ0; 0;��2;

0; 0; 0�, the following would be the J , U , ˙ and V matrices as depicted in (2.2):

J D

2

66666664

0 0 0 0 0 020 0 0 0 0 00 20 10 0 0 00 0 0 1 0 10 1 1 0 1 01 0 0 0 0 0

3

77777775

;

U D

2

66666664

0 0 0 0 1 �0:00340 �0:9988 0 0 �0:0002 �0:0499

�0:9982 0 0 0:06 0 00 0 �1 0 0 0

0:06 0 0 0:9982 0 00 �0:0499 0 0 0:0034 0:9987

3

77777775

;

34 A. Elkady et al.

˙ D

2

66666666664

22:401 0 0 0 0 0

0 20:025 0 0 0 0

0 0 1:4142 0 0 0

0 0 0 1:0935 0 0

0 0 0 0 0 0

0 0 0 0 0 0

3

77777777775

;

V D

2

6666666664

0 �1 0 0 0 0

�0:8939 0 0 0:1852 0:4074 �0:027

�0:4483 0 0 �0:3638 �0:8147 0:0539

0 0 �0:7071 0 �0:0467 �0:7056�0:0027 0 0 �0:9129 0:4074 �0:027

0 0 �0:7071 0 0:0467 0:7056

3

7777777775

:

It is obvious that in the singular matrix ˙ , �5 and �6 assume the value zerowith small tolerance. This is due to the fact that there are two singular cases in itsconfiguration; the fourth and sixth joints are on same axis and it is in a singular armconfiguration, and thus, �5 is zero.

2.3.5 Proposed Manipulability Measure Algorithm

To justify the proposed method, the following algorithm is proposed:

Algorithm 1: Calculate the manipulability index1: Find the joint(s) that may lead to a singular configuration assuming that the number of these

jointsD n.2: for i D 1 to n do3: Change the value of the i th joint from its initial to its final value using simulation

software – Matlab robotic toolbox [2] is used in our case.4: Calculate the Jacobian (J) and singular (˙ ) matrix.5: Plot every normalized � and also the rank of the Jacobian matrix.

Normalized�i D�i

Maxf�i1;�i2; �i3; ::::::; �ing(2.7)

Where: i is the order of the � in the singular matrix and n is the number of steps during thesimulation.

6: Check the rank of the Jacobian matrix.7: end for


2.4 Experiments

In this section, we will show and explain some results using serial manipulators withDH parameters illustrated in Tables 2.1–2.3. We have proposed some assumptionswhich can be summarized as follows:

� In our case study, we have dealt with the arm manipulability regardless of theorientation singularity.

� We study non-redundant manipulators only.

2.4.1 The Puma 560 Manipulator

In the Puma 560, we have experienced that the third joint is the cause of singu-larity. The sample trajectory of this manipulator from the initial position Qinitial D

Œ0; 0;��2 ; 0; 0; 0� to the final position Qfinal D Œ0; 0; �2 ; 0; 0; 0� is shown in Fig. 2.1.The DH parameters of the Puma 560 are shown in Table 2.1.

Table 2.1 DH parameters of the Puma 560 manipulatori ˛ a d Initial limit Final limit Joint’s type1 90 0 * 0 �170 170 R2 0 0.4318 * 0 �225 45 R3 �90 0.0203 * 0.15005 �250 75 R4 90 0 * 0.4318 �135 100 R5 �90 0 * 0 �100 100 R6 0 0 * 0 �180 180 R

Table 2.2 DH parameters of a six degrees of freedom (DOF) serialmanipulatori ˛ a d Initial limit Final limit Joint’s type1 90 0 * 10 �170 170 R2 0 10 * 0 �225 45 R3 �90 0 * 0 �250 75 R4 90 0 * 10 �135 100 R5 �90 0 * 0 �100 100 R6 0 0 * 0 �180 180 R

Table 2.3 Manipulability’s bands of the Mitsubishi Movemaster manipulatorin 2D workspacei ˛ a d Initial limit Final limit Joint’s type1 90 0 * 300 �150 150 R2 0 250 * 0 100 130 R3 0 160 * 0 �110 0 R4 �90 0 * 0 �90 90 R5 0 0 * 72 0 0 R

36 A. Elkady et al.

Fig. 2.1 Phases for the Puma560 manipulator, changingfrom the initial singularconfiguration to the finalsingular configuration withthe corresponding rank


Fig. 2.2 The behaviour of �1 to �6 during the experiment

In Fig. 2.2, it is obvious that �5 is exactly expressing the manipulability’s index.Furthermore, the rank of the Jacobian matrix during this experiment was constantat 5 because joints 6 and 4 were on same axis during the whole experiment. Themanipulability index of every point within the whole workspace is represented inbands and each band is visualized using a different color as shown in Fig. 2.3.

Figure 2.3 is considered important in our research strategy since it providesa visual demonstration for the manipulability measure for the entire workspacefor the Puma 560 manipulator. These results can strongly contribute in develop-ing an intelligent mobile manipulator. For example, the positions with the highestmanipulability index will have better dexterity compared with those of the lowestmanipulability index.

2.4.2 A Six Degrees of Freedom Serial Manipulator

Similarly, we implemented the same procedure for the Mitsubishi Movemaster ma-nipulator and a regular six DOF manipulator. In a six DOF manipulator, the sample

38 A. Elkady et al.

Fig. 2.3 Manipulability’s bands of the puma 560 manipulator in 2D workspace according to �5

trajectory of this manipulator is from the initial position Qinitial D Œ0; 0;��2 ; 0; 0; 0�

to the final position Qfinal D Œ0; 0;�2; 0; 0; 0�. The DH parameters of this manipula-

tor are shown in Table 2.2. The behaviour of �3 during the experiment is shown inFig. 2.4. The visual demonstration of the manipulability index is shown in Fig. 2.5.

2.4.3 The Mitsubishi Movemaster Manipulator

The initial position is Qinitial D Œ0; 0;��2 ; 0; 0; 0� and the final position Qfinal D

Œ0; 0; �2 ; 0; 0; 0�. The DH parameters of this manipulator are shown in Table 2.3.The behaviour of �3 during the experiment is shown in Fig. 2.6. The visual demon-stration of the manipulability index is shown in Fig. 2.7.

2.4.4 Experimental Results

It is obvious from Table 2.4 that we can suppose that the order of � that is expressingthe kinematics manipulability’s index equals to the minimum rank of the Jacobianmatrix.


Fig. 2.4 The behaviour of �3 during the experiment

Fig. 2.5 Manipulability’s bands of a six degrees of freedom (DOF) manipulator 2D

40 A. Elkady et al.

Fig. 2.6 The behaviour of �3 during the experiment

Fig. 2.7 Manipulability’s bands of the Mitsubishi Movemaster manipulator in 2D workspace


Table 2.4 Summary of resultsOrder of � expressing Min rank of the

Manipulator the manipulability Jacobian matrixPuma 560 5 5Six DOF 3 3Mitsubishi movemaster 3 3

2.5 Mobile Manipulator

A mobile manipulator is a manipulator mounted on a mobile platform with no sup-port from the ground. A mobile manipulator offers a dual advantage of mobilityoffered by the platform and dexterity offered by the manipulator. For instance, themobile platform extends the workspace of the manipulator. The DOF of the mo-bile platform also adds to the redundancy of the system. The mobile manipulatoris required to perform complicated tasks and is potentially useful in dangerous andunpredictable environments, such as at a construction site, space, underwater, con-struction, service environments and in nuclear power station. Arai [1] suggested thatthe mobile manipulator system should be capable of both locomotion and manipu-lation when it is applied to various tasks in construction, agriculture, home, officeand hospital services. Then he described why and how locomotion and manipula-tion should be integrated, what the benefits are and what problems must be solvedin terms of practical application of the robot. In [9], a systematic modelling of thenonholonomic mobile manipulators is proposed. The derived models are used togeneralize the standard definition of manipulability to the case of mobile manipula-tors. In addition, the effects of mounting a robotic arm on a nonholonomic platformwere shown through the analysis of the manipulability.

In fixed based serial manipulators, manipulability depends on link lengths, jointtypes, joint motion limits and the structure of the manipulator. In mobile manipu-lator, the manipulability depends on the kinematics, geometric design, the payloadand mass and mass distribution of the mobile platform. Thus, the manipulabilitymeasure in mobile manipulators is very complicated due to the coupling betweenthe kinematic parameters and the dynamics effect.

We extend the manipulability concept commonly used for serial manipulatorsto general mobile manipulator systems. To study the manipulability and dexteritymeasure for a mobile manipulator, we first study those of the fixed base serial ma-nipulator as discussed in the previous sections.

2.6 RISC Mobile Manipulator

We are developing and constructing the mobile manipulator platform RISC. Theprototype of the RISC is shown in Fig. 2.8. The RISC mobile manipulator has beendesigned to support our research in algorithms and control for autonomous mobile

42 A. Elkady et al.

Fig. 2.8 A prototype of the RISC manipulator

manipulator. The objective is to build a hardware platform with redundant kine-matic DOF, a comprehensive sensor suite and significant end-effector capabilitiesfor manipulation. The RISC platform differs from any related robotic platforms be-cause its mobile platform is a wheelchair base. Thus, the RISC has the advantagesof the wheelchair, such as high payload, high speed motor package (the top speed ofthe wheelchair is 6 miles/h), Active-Trac and rear caster suspension for outstandingoutdoor performance and adjustable front anti-tips to meet terrain challenges.

2.7 Modelling of the RISC

2.7.1 The Position of the Robot

In order to specify the position of the robot on the plane, we establish a relationshipbetween the global reference frame of the plane and the local reference frame ofthe robot. The origin O of the global reference frame is selected at arbitrary on theplane as shown in Fig. 2.9. The point C is the centre of mass of the robot. The originP of the local reference frame of the robot fXp ,Ypg is at the centre of the robot.The basis defines two axes relative to P on the robot chassis and is, thus, the robot’slocal reference frame. The position of P in the global reference frame is specified bycoordinates x and y and the angular difference between the global and local referenceframes is given by . The pose of the robot is described by a vector .

D

2

4x

y

3

5 : (2.8)


Fig. 2.9 Kinematic model of the RISC

To describe the robot’s motion, it will be necessary to map the motion alongthe axes from the global reference frame to the robot’s local reference frame. Thismapping is accomplished using the orthogonal rotation matrix R./, where

R./ D

2

4cos./ sin./ 0�sin./ cos./ 00 0 1

3

5 :

This orthogonal rotationR./ is used to map the motion in the global referenceframe to motion P in terms of the local reference frame fXp ,Ypg. This operation is:

P D R./ (2.9)

2.7.2 The velocity of the robot

Given that the spinning speed of left wheel is P̊l and the spinning speed of the rightwheel is P̊r , a forward kinematic model would predict the robot’s overall speed inthe global reference frame. The linear velocity of the centre of the right wheel is vr ,where

vr D r P̊r : (2.10)

In addition, the velocity of the left one is vl where

vl D r P̊l : (2.11)

44 A. Elkady et al.

Using the instantaneous centre of rotation (ICR), if the linear velocity of the centreP of the robot is V and its angular velocity !, we can find that:

V Dvr C vl2

; (2.12)

! D P Dvr � vl2L

: (2.13)

The velocity of point P in its local reference frame P Pp:

P Pp D

2

64

V

0

P

3

75 : (2.14)

The velocity of point C in the local reference frame fXp ,Ypg is P Pc where

P Pc D

2

64

V

Pd

P

3

75 : (2.15)

The velocity of point P in the global reference frame Pp is

Pp D R./�1P Pp; (2.16)

Pp Dr

2

2

64

. P̊l C P̊r / cos./

. P̊l C P̊r / sin./. P̊r� P̊l /

l

3

75 : (2.17)

The velocity of point C in the global reference frame Pc is

Pc D R./�1P Pc : (2.18)

Thus,

Pc D

2

664

Pxc

Pyc

P

3

775 D

r

2

2

664

. P̊l C P̊r / cos./C dl. P̊l � P̊r / sin./

r. P̊l C P̊r / sin./C dl. P̊r � P̊l / cos./

. P̊r� P̊l /l

3

775 : (2.19)


2.7.3 Kinematic Constraints

However, several important assumptions will simplify this representation. First, theplane of the wheel always remains vertical and that there is, in all cases, one singlepoint of contact between the wheel and the ground plane. Furthermore, there is nosliding at this single point of contact, so the wheel undergoes motion only underconditions of pure rolling and rotation about the vertical axis through the contactpoint.

The fixed standard wheel has no vertical axis of rotation for steering. Its angle tothe chassis is fixed and it is limited to back and forth motion along the wheel planeand rotation around its contact point with the ground plane. The rolling constraintfor this wheel enforces that all motions along the direction of the wheel plane mustbe accompanied by the appropriate amount of wheel spin. Thus, the first constraint is

Pyc cos./ � Pxc sin./ � Pd D 0: (2.20)

Furthermore, there are two rolling constraints, i.e., the driving wheels do not slip,

Pxc cos./C Pyc sin./C Pl D r P̊r ; (2.21)Pxc cos./C Pyc sin./ � Pl D r P̊l : (2.22)

Letting q D Œxcyc ; ; ˚r ; ˚l �t , the three constraints can be written in the form of:

A.q/ Pq D 0; (2.23)

where

A.q/ D

2

4sin �cos d 0 0

cos sin l �r 0

�cos �sin l 0 r

3

5 and Pq D

2

666664

PxcPycPP�rP�l

3

777775

:

The last two equations are called the nonholonomic constraint because they are notintegrable differential equations.

2.8 Conclusions and Future Work

In this chapter, we present a new algorithm for measuring manipulability, andthen we implement simulations supporting our methodology on different manipula-tors. We describe how mobile manipulator capabilities are a key to several roboticapplications and how the manipulability measure is crucial in performing intelli-gent behaviour tasks such as grasping, pulling or pushing objects with sufficientdexterity.

46 A. Elkady et al.

In our anticipated future work, there will be an ongoing effort for the develop-ment of multiple mobile manipulator systems and platforms, which will interactwith each other to perform more complex tasks exhibiting intelligent behavioursutilizing the proposed manipulability measure.

References

1. T. Arai (1996) Robots with integrated locomotion and manipulation and their future. Proceed-ings of the 1996 IEEE/RSJ International Conference on Robots and Intelligent Systems, IROS96, vol. 2, Osaka, pp. 541–545.

2. P.I. Corke (2002) Robotic toolbox.3. C.M. Gosselin (1990) Dexterity indices for planar and spatial robotic manipulators. Pro-

ceedings of the International Conference on Robotics and Automation, vol. 1, May 1990,pp. 650–655.

4. C. Gosselin, J. Angeles (1988) The optimum kinematic design of a planar three-degree-of-freedom parallel manipulator. Journal of Mechanisms, Transmissions and Automation inDesign 110, 35–41.

5. K. Gotlih, I. Troch (2004) Base Invariance of the Manipulability Index. Robotica archive. Cam-bridge University Press, New York, NY, USA, 22(4), pp. 455–462.

6. C.A. Klein, B.E. Blaho (1987) Dexterity measures for the design and control of kinematicallyredundant manipulators. The International Journal of Robotics Research 6(2), 72–83.

7. J. Lee (1997) A study on the manipulability measures for robot manipulators. IntelligentRobots and Systems 3(7–11), 1458–1465.

8. J.K. Salisbury, J.J. Craig (1982) Articulated hands. Force control and kinematic issues. TheInternational Journal of Robotics Research 1(1), 4–17.

9. R. Siegwart, I.R. Nourbakhsh (2004) Intelligent Robotics and Autonomous Agents Series. MITPress, Cambridge, MA, ISBN 0-262-19502-X.

10. T.M. Sobh, D.Y. Toundykov (2004) Optimizing the tasks at hand. IEEE Robotics and Automa-tion Magazine 11(2), 78–85.

11. T. Tanev, B. Stoyanov (2000) On the Performance Indexes for Robot Manipulators. Problemsof Engineering Cybernetics and Robotics.

12. K. van den Doel, D.K. Pai (1996) Performance measures for robot manipulators: A unifiedapproach. The International Journal of Robotics Research 15(1), 92–111.

13. T. Yoshikawa (1985) Manipulability of robotic mechanisms. The International Journal ofRobotics Research 4(2), 3–9.

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